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Duration dependence affects the dynamics of multi sate time to event outcomes. In this paper we are testing if a contraction or an expansion state for the housing price is duration dependent on previous states lengths. This test has implications for explaining the dynamics and the predictability of the housing prices in subsequent spells of contraction/expansion. The test is carried on using a discrete time duration model. This research shows that federal fund rate has strong effect on duration of both expansion and contraction. The analysis is also showing that while for both contraction and expansion spells we observe duration dependence, the risk of exiting from either spell at the beginning of the spell is practically flat for the first five to six years in the expansion spells and between seven and eight years in the contraction spells. After these periods the risk of exiting an expansion spell is increasing but in a non-monotone way, while for the contraction spell the risk of exiting the state is increasing in a monotone way, making the contraction periods easier to predict than the expansion periods.

There is extensive work about the contribution of the housing market to the economic growth of a given economy. The importance of this sector for the US economy was pointed out by Belsky and Prakken [

As a consequence, after a period where we experienced both a sustainable expansion and an increase in the housing pricing at limits that were not sustainable by the market, the recent downturn was abrupt with serious consequences for the economies dependent on the development of this sector and to the world economy in general.

An important question about housing price dynamics and their predictability is whether the length of a contraction or an expansion state is duration dependent on previous states lengths. In this regard, positive duration dependence implies an increase in the probability of an exit as the time passes, while negative duration dependence means a decrease in the probability of an exit with time. Therefore, duration dependence, if present in a market and, if properly identified can be used to make predictions about possible turning points in the housing prices. Alternatively, if there is no duration dependence, or in other words turning points in prices are independent on the length of a given state, it becomes hard to make predictions about the lengths of a given state. It becomes evident that to understand the duration dynamics of the housing market one needs to properly identify if duration dependence is present, and if it is present, to identify the type and the magnitude of this dependence.

There are many papers that investigate the factors that are important for the housing prices, some of them focusing on long run models for housing prices. Using data on owner-occupied housing cost in the US metro areas, Himmelberg et al. [

In this paper, a multiple spells duration model is used with the purpose to properly identify the presence and the type of the duration dependence for the prices in the housing market in both contraction and expansion cycles.

Duration dependence was previously identified in the housing market using different methods. Durland and McCurdy [

Mills [

The results of previous research were mixed about the issue of duration dependence. A natural way of identifying duration dependence is by using duration analysis, in particular hazard models, as they are flexible in specification and can account for different types of censoring in the data. Our analysis is showing that while for both contraction and expansion spells we observe duration dependence, the risk of exiting from either spell at the beginning of the spell is practically flat for the first five to six years in the expansion spells and between seven and eight years in the contraction spells. After these periods the risk of exiting an expansion spell is increasing but in a non-monotone way, while for the contraction spell the risk of exiting the state is increasing in a monotone way, making the contraction periods easier to predict than the expansion periods.

The paper is organized as follows. Section 2 discusses the methodology, Section 3 discusses the Data, Section 4 analyzes the results while Section 5 concludes the analysis.

This paper uses duration modeling to investigate whether there is duration dependence in housing price cycles or not. The type of duration model used is conditioned on the frequency and length of the available duration data. While continuous models are relevant for high frequency data, discrete duration models may be more appropriate for lower frequency durations. An additional advantage of using discrete duration modeling is that they are more tractable as it is easier to incorporate time-varying covariates.

Jenkins [

The following notation is used:

is

terminating event occurs. The distribution of our duration outcome variable is modeled using the probability of having a spell that ends at each

where

with its survival defined as:

Under the assumption of independent spells, the total likelihood generated by the discrete hazards can be expressed as:

where

Since

The corresponding log-likelihood function is:

which is similar to the standard log-likelihood function for a regression analysis with a binary variable

The expectation of the dichotomous event indicator

We have that

Thus, the values of the event indicator are observed realizations of hazard probability. The above probability model can be conditioned on observables of the form

The analysis is from the Office of Federal Housing Enterprise Oversight (OFHEO) housing price indices (HPI) and is based on an annual data collected for the largest 110 cities in the United States during the period 1975- 2007. However, for some cities the housing price index is available after 1975. Also, to deal with the potential left censoring of a given spell, the data is trimmed to include only complete spells. The adjustments made the panel to be unbalanced. Consumer Price Index (CPI) is used to transform the nominal data to real data, where CPI was collected from Bureau of Labor Statistics of United States. As covariates, the model includes the Effective Federal Funds Rate (EFFR), which is a good variable to capture the policy makers’ effect on the housing price. The source of these data is St. Louis Federal Reserve. Other fundamental economic variables that are entered in the model are Population Size (PS) and Personal Income (PI). The data at metro level was collected from the Bureau of Economic Analysis. Like other nominal variables in the model, the PI was deflated by CPI.

Fundamental variables are tested for stationarity. PS is stationary in level, while the Real PI and EFFR are not stationary in level but are stationary in the difference. To test PI and PS, Levin-Lin-Chu Unit root test for panel data is used (see

The estimation was done separately for the expansion and the contraction of the housing cycle spells. As there is no formal test for the type of unobserved heterogeneity, both fixed effects and random effects panel logit models are employed. Both models are consistent under the null of random effects, but if there are significant differences in the parameter estimates between the two models, the random effects estimates are biased. As the results of the random effects model show some differences when they are compared with the fixed effects model, the fixed effects panel logit results are chosen to be reported.

The parameter estimates of the expansion spells are presented in

Levin-Lin-Chu test | Maddalla-Wu test | ||||||
---|---|---|---|---|---|---|---|

Lags | t-value | t-star | P > t | Lags | Chi2 | P > Chi2 | |

Personal income | 1 | −23.158 | −10.349 | 0.0000 | 2 | 342.625 | 0.0000 |

Population | 0 | −17.454 | −6.362 | 0.0000 | 0 | 290.850 | 0.0000 |

Lags | DF Test statistics | 5% critical value | |
---|---|---|---|

Effective Fed Fund Rate | 1 | −4.199 | −3.576 |

Total | First spell | Second spell | Third spell | |
---|---|---|---|---|

Personal Income | −1.694 | −11.980^{**} | −9.710^{***} | 2.736 |

(2.295) | (5.220) | (5.457) | (9.867) | |

Population | −4.7E−08 | 0.000 | −6.7E−08 | 5.3E−08 |

(3.6E−08) | (0.000) | (5.6E−08) | (3.8E−07) | |

Interest Rate | 0.008^{*} | 0.012^{**} | 0.009^{*} | 0.012^{**} |

(0.002) | (0.005) | (0.002) | (0.005) | |

Second Period | −0.379 | 0.145 | −1.372^{**} | −0.351 |

(0.291) | (0.394) | (0.587) | (0.937) | |

Third Period | −0.754^{**} | −0.552 | −1.144^{**} | −0.109 |

(0.327) | (0.471) | (0.544) | (0.842) | |

Fourth Period | −0.832^{**} | −0.335 | −1.992^{*} | 0.232 |

(0.343) | (0.464) | (0.772) | (0.785) | |

Fifth Period | −0.335 | 0.696^{***} | −30.235 | −2.025 |

(0.307) | (0.404) | (900004) | (1.303) | |

Sixth Period | 0.063 | 1.202^{*} | −1.977^{*} | −0.419 |

(0.296) | (0.430) | (0.780) | (0.944) | |

Seventh Period | 0.730^{*} | 1.204^{**} | −0.292 | 2.189^{*} |

(0.278) | (0.508) | (0.457) | (0.747) | |

Eighth Period | −0.714^{***} | 0.651 | −1.552^{**} | −0.424 |

(0.416) | (0.667) | (0.637) | (1.298) | |

Ninth Period | 0.281 | −30.728 | 0.275 | 0.733 |

(0.338) | (2525436) | (0.435) | (0.995) | |

Tenth Period | 0.327 | −0.316 | 0.341 | −28.985 |

(0.373) | (0.929) | (0.469) | (1812799) | |

Eleventh Period | −0.266 | 0.304 | −0.446 | 0.287 |

(0.481) | (0.890) | (0.612) | (1.354) |

Twelfth Period | 0.289 | −29.396 | 0.065 | 0.918 |
---|---|---|---|---|

(0.427) | (2063138) | (0.560) | (1.301) | |

Thirteenth Period | 0.233 | −29.480 | 0.572 | −28.130 |

(0.474) | (2046891) | (0.561) | (1610780) | |

Fourteenth Period | 0.359 | −35.745 | 0.869 | −0.696 |

(0.509) | (3795794) | (0.669) | (1.421) | |

Fifteenth Period | 0.416 | −30.250 | 0.486 | 2.125 |

(0.615) | (3078385) | (0.899) | (1.358) | |

Sixteenth Period | 2.107^{*} | 1.559^{***} | 2.079^{**} | - |

(0.615) | (0.906) | (0.964) | - | |

Seventeenth Period | −30.874 | −31.169 | −28.410 | - |

(5008294) | (4465425) | (2648193) | - | |

Eighteenth Period | −29.763 | −29.448 | −28.711 | - |

(3203434) | (2908934) | (2648960) | - | |

Nineteenth Period | 1.250 | 0.700 | 1.287 | - |

(0.935) | (1.276) | (1.451) | - | |

Twentieth Period | −30.721 | −30.725 | −30.164 | - |

(4094580) | (3505588) | (3757681) | - | |

Constant | −1.603^{*} | −1.029^{*} | −1.223^{*} | −2.259^{*} |

(0.210) | (0.369) | (0.354) | (0.773) | |

Observation | 1679 | 574 | 858 | 221 |

Log Likelihood | −651.933 | −247.007 | −251.476 | −76.556385 |

Notes: Standard errors are in parentheses. ^{*}, ^{**}, and ^{***} denote 1%, 5%, and 10% level of significance, respectively.

spells, which means that PS and PI may have negative effect on probability of exiting from the expansion spells, whilst EFFR may have a positive effect on the exiting probability from the expansion spells. The opposite effects are recorded for the contraction spells (see

The dummy variables of each period are capturing the time effects or the baseline hazards patterns. The results for the expansion spells are to some extent mixed, which means that the risk of leaving an expansion spell does not have a stable pattern. The conditional (estimated) plots of first, second and third expansion spells are presented in

The plots suggest that the estimated hazards are mimicking very closely the empirical hazard nonparametric estimates, which suggest a very good fit obtained with the proposed conditional analysis. The graphs also suggest that within the three analyzed categories there are structural differences, with the first and third expansion spells having the highest hazard (probability of exiting the state in the next observed period) in the middle of the period, while for the second expansion spell the hazard is looking more exponential (increasing towards the end of the spell). The first and third spells are also showing increasing hazards towards the end of the spells, but not as high as the increase in the hazard for the second spell.

The conditional (estimated) plots of first, second and third contraction spells are presented in

The plots suggest that the estimated hazards are mimicking very closely the empirical hazard nonparametric estimates as in the case of the expansion spells, and in contractions periods the behavior of the price is similar across different spells. If we pool all the information form all the spells for the expansion and separately for the contraction spells we generate an equivalent of an average expansion/contraction spell.

Total | First spell | Second spell | Third spell | |
---|---|---|---|---|

Personal Income | 26.148^{*} | 29.051^{*} | 31.913^{*} | 2.925 |

(3.419) | (5.113) | (6.743) | (11.910) | |

Population | −1.1E−08 | 1.4E−07^{**} | −8.8E−08 | −3.6E−07 |

(3.7E−08) | (6.5E−08) | (5.9E−08) | (4.6E−07) | |

Interest Rate | −0.013^{*} | −0.026^{*} | −0.008 | 0.024^{**} |

(0.004) | (0.006) | (0.006) | (0.012) | |

Second Period | 1.263^{*} | 1.198^{**} | 1.577^{*} | −0.040 |

(0.305) | (0.568) | (0.465) | (0.720) | |

Third Period | 1.098^{*} | 1.620^{*} | 0.661 | −0.129 |

(0.325) | (0.564) | (0.580) | (0.779) | |

Fourth Period | 0.944^{*} | 1.568^{*} | 0.407 | −0.869 |

(0.351) | (0.580) | (0.671) | (1.166) | |

Fifth Period | 1.081^{*} | 1.094^{***} | 0.928 | 2.489 |

(0.349) | (0.607) | (0.591) | (0.900) | |

Sixth Period | 1.335^{*} | 2.133^{*} | 0.842 | 1.166 |

(0.364) | (0.606) | (0.611) | (1.404) | |

Seventh Period | 1.688^{*} | 1.745^{**} | 2.073^{*} | 1.208 |

(0.389) | (0.709) | (0.592) | (1.651) | |

Eighth Period | 1.896^{*} | 2.304^{*} | 2.057^{*} | −28.497 |

(0.427) | (0.725) | (0.676) | (2329017) | |

Ninth Period | 2.826^{*} | 3.147^{*} | 3.326^{*} | - |

(0.461) | (0.704) | (0.874) | - | |

Tenth Period | 1.293^{***} | 1.914^{**} | 1.187 | - |

(0.727) | (0.969) | (1.381) | - | |

Eleventh Period | 2.016^{*} | 2.623^{*} | 2.128 | - |

(0.722) | (0.917) | (1.601) | - | |

Twelfth Period | 2.866^{*} | 3.924^{*} | −25.722 | - |

(0.960) | (1.275) | (1708642) | - | |

Thirteenth Period | 2.377^{***} | - | −26.303 | - |

(1.445) | - | (1708642) | - | |

Fourteenth Period | −28.082 | - | −27.155 | - |

(2720641) | - | (1708642) | - | |

Constant | −3.084^{*} | −3.903^{*} | −3.012^{*} | −1.018 |

(0.281) | (0.533) | (0.460) | (0.734) | |

Observation | 1057 | 549 | 403 | 90 |

Log Likelihood | −480.876 | −224.068 | −170.641 | −44.929 |

Notes: Standard errors are in parentheses. ^{*}, ^{**}, and ^{***} denote 1%, 5%, and 10% level of significance, respectively.

empirical and estimated hazard results for the polled Expansion and Contraction Spells.

Again, both the empirical and the conditional estimated hazards are moving very closely to each other, suggesting a very good fit of the estimated model. We also see that the average expansion spell is more heterogeneous and longer than the average contraction spell. To look at the issue of duration dependence we followed the estimated year dummies. If there is to be no duration dependence, then the period specific dummies should not be significantly different, regardless of the duration of time spent in the spell. The fact that we observe significantly different period dummy variables ensures that there is duration dependence in both expansion and contraction

spells. The forms of duration dependence for both phases are mixed. While the pattern for a contraction spell is clearer. The risk of ending the spell declines between second and fourth year, while it increases after that, for the expansion spell we again observe more heterogeneity in the baseline hazard which suggest less evidence of duration dependence. Therefore, it becomes clearer that while contraction spells are easier to predict, the expansion spells are harder to predict.

The main goal of this research is testing duration dependence in housing price market using discrete time duration models. By transforming the main model to a binary discrete choice model a fixed effects panel logit is used for the estimation, while assuming independence between the contraction and expansion spells. This research shows that federal fund rate has strong effect on duration of both expansion and contraction. So policy makers can influence duration by this instrument. The analysis is also showing that while for both contraction and expansion spells we observe duration dependence, the risk of exiting from either spell at the beginning of the spell is practically flat for the first five to six years in the expansion spells and between seven and eight years in the contraction spells. After these periods the risk of exiting an expansion spell is increasing but in a non-monotone way, while for the contraction spell the risk of exiting the state is increasing in a monotone way, making the contraction periods easier to predict than the expansion periods.